#512671

order	512
nilpotency class	5
isoclinism	isoclinic class of #512669
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#127	C₂⁴
prime radical	#3500	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3500 ◁ G
LECS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

805 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 805 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,14,17,8)(4,9)(5,10)(6,12,28,11)(7,13)(15,42,45,29)(16,43,47,30)(18,31,27,32)(19,48,49,57)(20,33)(21,35)(22,34,51,41)(23,52,53,36)(24,37)(25,39,55,38)(26,40)(44,61,63,56)(46,62,64,58)(50,59,54,60);(1,3,7,18)(2,8,13,32)(4,19,21,15)(5,23,26,16)(6,27,28,17)(9,29,35,57)(10,36,40,30)(11,31,12,14)(20,44,24,46)(22,45,51,49)(25,47,55,53)(33,58,37,56)(34,42,41,48)(38,43,39,52)(50,64,54,63)(59,62,60,61);(1,4)(2,9)(3,15)(5,24)(6,22)(7,21)(8,29)(10,37)(11,41)(12,34)(13,35)(14,42)(16,46)(17,45)(18,19)(20,26)(23,44)(25,54)(27,49)(28,51)(30,58)(31,48)(32,57)(33,40)(36,56)(38,59)(39,60)(43,62)(47,64)(50,55)(52,61)(53,63);(1,5,7,26)(2,10,13,40)(3,16,18,23)(4,20,21,24)(6,25,28,55)(8,30,32,36)(9,33,35,37)(11,38,12,39)(14,43,31,52)(15,44,19,46)(17,47,27,53)(22,50,51,54)(29,56,57,58)(34,59,41,60)(42,61,48,62)(45,63,49,64)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=hi,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ehi,f^b=fgh,g^b=gi,d^c=di,e^c=ei,f^e=fhi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317784, #72318941
maximal normal subgroup of	#72317784, #72318941

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419867);